I assume from your question you want the essential parts of an Arduino along with your circuit in a permanent form.
Here are my steps:
- Build the entire circuit on a breadboard or two. Then you know it all works.
- Transfer it to a permanent form of circuit, testing as you go.
Take it a step at a time, and it'll work out nicely.
1. Build the entire circuit on a breadboard or two.
Then you know it all works, and that you have all of the components ready to make your permanent circuit
This will involve making and programming part of your Arduino on the breadboard. I don't think you'll need the USB part, so it can be through-hole components which are relatively straightforward to begin with.
Here is a link to instructions which show how to make an Arduino without USB.
It can be programmed using your existing Arduino, you don't need a programmer. For example:
I'd make a sketch of the circuit schematic to ensure I understood what I had on the breadboard, and to guide the next step.
2. I'd recommend you transfer the whole thing to 'veroboard' (stripboard) rather than make a PCB.
Veroboard/stripboard will be much quicker and cheaper. You could easily spend 10x longer learning to use Eagle well enough to get a PCB made, than it would take to design and make the entire circuit on veroboard/stripboard.
You can do a design on squared paper, but there are some CAD programs to help if you google for veroboard CAD. I have never used them, I use paper and a soft pencil or a vector drawing package. A friend used PowerPoint because that's what he had to hand.
Here is an example of someone who has built an Arduino on veroboard/stripboard.
It shows the design he produced for veroboard/stripboard. Try to get a reasonable layout for your design before making it. This is where the soft pencil and eraser come in :-)
Typically the first couple of attempts are too big or too small. Make it easy for yourself and get plenty of squared of graph paper :-)
This link shows what the process will look like.
You can follow veroboard/stripboard arduino design, and test that it works. Then focus on your extra circuit.
Use a socket for the microcontroller, and so don't solder it in directly. Most of the other parts are a few $'s total, so I'd get a few of each part for spares and practice (To put it in context, some electronics companies charge more for delivery than those parts will cost, so getting several sets of parts makes sense, especially if you intend to make some more things).
Total cost for the Arduino part should be under $10.
Good luck, and I hope you enjoy it.